Luminosity Scans for Beam Diagnostics

A new type of fast luminosity separation scans ("Emittance Scans") was introduced at the CERN Large Hadron Collider (LHC) in 2015. The scans were performed systematically in every fill with full-intensity beams in physics production conditions at the Interaction Point (IP) of the Compact Muon Solenoid (CMS) experiment. They provide both emittance and closed orbit measurements at a bunch-by-bunch level. The precise measurement of beam-beam closed orbit differences allowed a direct, quantitative observation of long-range beam-beam PACMAN effects, which agrees well with numerical simulations from an improved version of the TRAIN code.


I. INTRODUCTION
The transverse beam emittance is a key parameter defining the luminosity in a collider. Complementary to existing beam instrumentation, the convoluted beam width at an interaction point, where the two beams collide, can be measured by separating the beams and observing the change in luminosity. This technique was pioneered by Simon van der Meer [1] at the CERN Intersecting Storage Rings, and is commonly used for calibrating the experiment's luminosity monitors.
In the following, we present a scheme for luminosity scans called "Emittance Scans" which can be performed in regular physics production conditions within a short time. Analyzing the bunch-by-bunch luminosity measurements during these scans allows deriving transverse emittances and closed orbit offsets at a bunch-by-bunch level.
Emittance Scans were first commissioned at the LHC in 2015 and have been done regularly ever since. As they depend only on the linearity of the luminosity measurements and the transfer function of the used steering magnets, they are fully independent of any beam instrumentation commonly used for transverse emittance measurements and the scans provide important complementary data. Examples are the quantification of the transverse emittance optimizations in the LHC injector complex in 2015 or for understanding the observed asymmetry in delivered luminosities between the two high-luminosity experiments ATLAS and CMS in 2016.
In this article, we present the methodology and results from Emittance Scans performed in 2015-2017 LHC operation. In Sec. II, we give an overview of the operational setup at the LHC and the resulting beam-beam effects. The methodology and analysis of emittance scans is described in Secs. III-VI. Finally, we present the measured emittances at the LHC in Sec. VII and compare them to complementary measurements * michi.hostettler@cern.ch from beam instrumentation. The measured closed orbit separations at the interaction points are shown in Sec. VIII and compared to simulations; there, we also discuss the recent improvements to the TRAIN simulation code to accurately predict bunch-by-bunch beam-beam effects at the LHC.

II. BEAM-BEAM ORBIT EFFECTS AT THE LHC
For physics production at the LHC, each of the two rings is typically filled with more than 2000 bunches [2]. The bunches are mostly spaced by 25 ns, but longer gaps separate them in "trains" of e.g. 96 or 144 bunches to allow for injection and extraction kicker rise times both in the injectors and the LHC. Certain effects and systems affect the bunches differently depending on their position in the train. For example, electron-cloud effects generally degrade the beam parameters towards the end of the train. Also, as explained in the following, the number of parasitic long-range beam-beam encounters depends on the position in the train. Hence, bunchby-bunch beam diagnostics are crucial to optimize the performance of the LHC.
When the two beams share a common beam pipe around the 4 LHC Interaction Points (IPs), parasitic collisions outside of these points are avoided by crossing the beams at an angle. The plane in which this crossing angle is applied is referred to as the "crossing plane" (vertical for ATLAS/IP1, horizontal for CMS/IP5), while the orthogonal plane is referred to as the "separation plane". Even though direct parasitic collisions are avoided by the crossing angle, the bunches still are affected by the fields of the other beam. The functional shape of this beam-beam force is shown in Fig. 1 range encounters in the drift space and the inner triplets, with a beam separation given by the crossing angle of the closed orbit (vertical in ATLAS, horizontal in CMS). Thereafter, the design orbits of the two beams are separated horizontally by the separation and the recombination dipoles over ∼108 m by 19.4 cm [2]. Due to the gaps between the bunch trains in the LHC filling schemes [3], bunches at the beginning and at the end of each bunch train are missing some long-range beam-beam encounters. This leads to a different net kick on these bunches, and hence, to bunch-to-bunch differences in orbit, tunes, and chromaticities. These so-called "PACMAN" effects have been predicted first in design studies of the SSC [4,5], and have since been observed in the Tevatron [6] and in the LHC [3,[7][8][9]. Also, due to the asymmetric position of the ALICE and LHCb experiments in the LHC, some bunches do not collide at all in these IPs ("SuperPACMAN" bunches).
While not hindering LHC operation, these PACMAN effects can affect the beam life time, and lead to bunch-to-bunch differences in orbit, tune and chromaticity, which can not be mitigated by global correctors (which do not act on individual bunches). The differences in orbit lead to bunch-to-bunch differences in the separation of the two beams at the IP, which can be measured through emittance scans. Despite of the small impact on the overall luminosity of these separations, they are a good benchmark for the overall long-range beambeam effects and PACMAN patterns in the machine.

III. EMITTANCE SCANS FOR GAUSSIAN BUNCHES
For Gaussian Beams, in the presence of a beam separation, the luminosity of a colliding bunch pair is given by [10] where N 1,2 are the bunch intensities, f rev is the revolution frequency, and Σ x , Σ y are the convoluted beam sizes in the x, y plane (including the effect of the crossing angle in the crossing plane).
In the presence of a separation d, the separation factor S is given by where Σ d is the convoluted beam size in the plane in which the separation d is applied. By scanning the separation d in each plane and fitting a Gaussian of the form to the measured luminosity, both the convoluted beam sizes Σ x,y and the initial (parasitic) separations d x0,y0 , can be determined. An example luminosity measurement during a scan with 7 separation steps is shown in Fig. 2, and the resulting fit is given in Fig. 3. If a bunch-by-bunch luminosity measurement is available, the fitting can be done independently for each colliding bunch pair.  To derive the transverse emittances ε x,y from the convoluted beam sizes Σ x,y , it is assumed that the beam sizes of Beam 1 and Beam 2 are equal 1 . In this case, the transverse beam size σ sep in the separation plane is given by In the crossing plane, the longitudinal bunch profile is projected onto the convoluted beam size Σ xing through the crossing angle. For a Gaussian longitudinal profile with bunch length σ z and half crossing angle α, the convoluted beam size is given by The normalized transverse emittances ε x,y is then derived from the transverse beam sizes σ x,y assuming a negligible dispersion at the IPs 2 and β * = β * x = β * y : where γ is the relativistic gamma factor and β * is the value of the β-function at the IP.

IV. EMITTANCE SCAN PARAMETERS
The scan parameters, namely the separation steps to scan and the time to acquire data at each step, were optimized to provide an adequate measurement while keeping the scans as short as possible. The steps were chosen to be equidistant and symmetric with respect to the initial position of the beams. A total effective separation of ∼3σ is aimed for, corresponding to a separation factor of S ≈ 0.1 and hence a change in luminosity by a factor of ∼10. Half of this total separation is applied symmetrically to the orbit of both beams with an opposite sign. Due to the crossing angle, Σ xing is larger and the effect of the separation is smaller in the crossing plane. This can be accounted for by applying a larger separation in the crossing plane.
The operational scan parameters established for the LHC based on these constraints are given in Table I. The scan range is given in units of the "LHC nominal" beam size σ nom (assuming a nominal emittance of ε nom = 3.5 µm), while the actual beam emittance was ε ≈ 2.5 µm as of 2016 [12]. Scans were done regularly at the CMS experiment as the experiment could deliver bunch-by-bunch luminosity readings at a high rate. A scan with these parameters reduces the luminosity at the scanned experiment for ∼1 min per plane. In the nominal LHC cycle, the bunch length is increased artificially during the energy ramp by injecting RF phase noise to guarantee beam stability [13]. This changes the longitudinal bunch profile from a Gaussian to a distribution with a larger core and lighter tails, which can be represented as a q-Gaussian with q ≈ 0.85 [14]. Over the course of several hours in collisions, the longitudinal distribution resumes a more Gaussian shape (Fig. 4). If the longitudinal distribution is not Gaussian, the simple factorization of the convoluted beam size in Eqn. 5 is no longer valid for the crossing plane [15]. While still assuming a Gaussian distribution in the transverse dimensions, the luminosity during a scan of the separation d in the crossing plane fulfills where α is the half crossing angle, f 1,2 are the longitudinal distribution functions for beam 1 and beam 2 respectively, and * denotes a convolution.
It is worth noting that α = 0 yields C(d) = const. and the separation dependency reduces to the separation factor as shown in Eqs. 2, 4. This also proves that the longitudinal distribution only affects scans in the crossing plane.
If the longitudinal distribution is measured for both beams, the convolution and the integration in Eq. 8 can be done numerically, and a non-linear regression of Eq. 7 to the measured scan points yields σ xing .

A. Simulations
To benchmark this approach, emittance scans were simulated for bunches with a transverse Gaussian and longitudinal q-Gaussian distribution with q = 0.85. This longitudinal distribution is typical for bunches in the LHC at the start of collisions. The simulations were done for two different sets of machine parameters and emittances, corresponding to the configurations used during the LHC 2015 and 2016 run periods. The simulated data was then analyzed using the method proposed above, and compared to the Gaussian approximation based on full-width half-maximum (FWHM) and true r.m.s. bunch length measurements. The operational bunch length measurement in the LHC is based on the FWHM method.
Results are compiled in Table II. While commissioning emittance scans during 2015 LHC operation, the Gaussian approach using the operational FWHM bunch length measurement was used and a systematic error of up to 15% in the crossing plane due to the non-Gaussian longitudinal distribution was assumed [16].
For measuring low-emittance beams at a large crossing angle, as at the LHC in 2016 and 2017, using measured longitudinal bunch profiles or a true r.m.s. bunch length measurement becomes crucial. For the machine parameters considered, the error on the emittance measurement using a Gaussian fit with a true r.m.s. bunch length measurement is below 5 %.

B. LHC Machine Studies
The impact of a changing longitudinal distribution on emittance scan was directly probed in 2016 during the commissioning of the longitudinal bunch flattening at the LHC [17].
During these tests, a sinusoidal RF phase modulation was applied to compensate for the shrinking caused by synchrotron radiation damping by flattening the bunch profiles. The distribution was measured before and after, the change is shown in Fig. 6.
Before and after the longitudinal bunch flattening, an emittance scan was done in the crossing plane. Results are shown in Fig. 7, and compared to the yield of Eq. 7 considering the  change in longitudinal distribution and intensity, but assuming no change in transverse emittance. The agreement shows that the longitudinal distribution is well accounted for.

C. Retrospective Correction
An operational measurement of the longitudinal bunch profiles was not available at the LHC as of 2017; longitudinal bunch profiles were only be logged by manual intervention on demand. However, since the machine parameters and the longitudinal distribution were reproducible, the difference between the reconstruction based on a FWHM measurement and the real emittance reduces to a constant offset at first order. This allows using the operational FWHM bunch length measurement and correcting the resulting emittance by an offset to be determined by a reference profile measurement and numerical simulations.
For emittance scans at the start of collisions in 2016, a simulation yields a correction of +0.5 ± 0.1 µm for the operational machine parameters and a typical bunch length of σ z = 7.8 ± 0.5 cm (Fig. 8). As shown in Fig. 9, this correction is only weakly dependent on the actual bunch length. Neglecting the bunch length dependence of the correction, the residual error on the emittance is less than 5 % for the range of bunch lengths used at the LHC.

VI. ERROR CONSIDERATIONS
At high luminosity, systematic errors are largely predominant, while statistical errors on the luminosity measurement are negligible. The error sources considered and their contribution to the total error of the derived emittances are given in Table III. For the crossing plane, it is assumed that the longitudinal bunch shape is taken into account up to a residual error of 5 %. It is to be noted that most systematic error sources only affect the absolute scales of the derived beam sizes and emittances. Relative bunch-to-bunch differences and other fit parameters (in particular the initial separation d 0 ) are not affected. Also, only the luminosity non-linearity due to pileup effects and the longitudinal bunch shape are expected to change over the course of a fill, and therefore to possibly affect the time evolution of the emittance.

A. Convoluted Emittances
The convoluted transverse emittances ε c = √ ε x ε y at the start of collisions, averaged over all bunches in the LHC, are shown Figs. 10,11 for the LHC proton physics runs in 2015 and 2016 respectively. For comparison, the other emittance measurements available for high-intensity beams are also displayed: the synchrotron light monitors [19] and the reconstruction from the luminosities measured by the ATLAS and CMS experiments. The available measurements agree better than 20 % in convoluted emittance, which is the expected level of systematic uncertainties [12].
In 2016 as of fill 5079, the convoluted emittance was reduced from 3.4 ± 0.6 µm to 2.0 ± 0.3 µm at the start of collisions with the introduction of the "Batch Compression Merging and Splitting" [20] beam production scheme in the injector complex. The resulting reduction in emittance is confirmed by all monitors.

B. Horizontal-Vertical Emittance Asymmetry
A difference between the ATLAS and CMS luminosities has been observed throughout 2016 LHC operation, while ideally the two experiments would receive the same luminosity from the LHC. As the beams cross in different planes in the two experiments, non-round beams can cause such an imbalance via the crossing angle (see Eqns. 4, 5) [21].
Non-round beams were indeed observed in emittance scans in 2016, as shown in Fig. 12. The larger emittance in the horizontal plane penalizes the ATLAS experiment (crossing vertically) less than the CMS experiment (crossing horizontally). The level of the measured emittance asymmetry is consistent with the ratio of the ATLAS and CMS luminosities [21].

C. Bunch-by-Bunch Emittances
Bunch-by-bunch emittances for a typical LHC fill are shown for all bunches in Fig. 13 and details for 6 typical bunch trains in Fig. 14. An increase in emittance over the individual trains by ∼0.5 µm is visible, attributed to electron cloud effects. Also, a decreasing trend in emittance is visible over the full ring. Bunches early in the ring are injected first, and thus circulate for a longer time in the LHC at injection energy where the degradation due to intra-beam scattering is strongest [22].  The measurements from the synchrotron light monitors and the emittance scans agree well within the expected systematic and statistical uncertainties. As expanded in section VI, the systematic uncertainties affecting the absolute emittance scale for all bunches are strongly dominant over per-bunch statistical or systematic errors.

VIII. CLOSED ORBIT SEPARATION MEASUREMENTS
For 2017 proton physics operation, PS batches of 48 bunches were joined to form trains of 96 bunches (2 PS batches) or 144 bunches (3 PS batches). The LHC filling scheme consisted of 15 trains with 144 bunches, interleaved with 4 trains of 96 bunches. Together with 12 non-colliding "witness bunches", this totaled to 2556 bunches per LHC beam (Fig. 15). The number of long-range beam-beam encounters per bunch in the drift space around the ATLAS and CMS experiments for this filling scheme is shown in Fig. 16. Due to PACMAN effects, the different number of beam-beam long-range encounters is expected to yield a difference in closed orbits, which can be observed using emittance scans.

A. Simulations
The TRAIN code [23,24] can simulate self-consistent closed orbits on a bunch-by-bunch level under the presence of long-range beam-beam effects. These simulations take into account the filling scheme, and optionally measured intensities and emittances. The simulation gives the bunch-by-bunch closed orbit for either beam, as well as bunch-by-bunch tunes and chromaticities. The difference between the closed orbits of the two beams at the IP yields to a separation, which can be observed during emittance scans. This effect has previously also been observed at the LHC during van-der-Meer scans [25].
In earlier versions, the TRAIN simulation only took into account the 15 long-range encounters in the IP drift space and inner triplets [24]. This approximation was justified for the LHC nominal design optics, but for the "achromatic telescopic squeeze" (ATS, [26]) optics used in the recent years of LHC operation, the effect of the long-range encounters beyond the 15th, while small, could sum up to non-negligible orbit kicks. This led to a discrepancy between measurements and simulation results in earlier studies, which were most pronounced in the horizontal (crossing) plane at CMS [27]. For this study, we therefore extended the TRAIN code to support an arbitrary number of long-range beam-beam encounters, taking into account both the closed and the design orbit. As shown in Fig. 17, considering the full 45 long-range encounters approximately doubles the PACMAN orbit effects in the horizontal plane at CMS, and introduces an additional fine structure over the bunch trains. A convergence study considering different numbers of long-range beam-beam encounters showed that at least 35-40 encounters per IP-side are needed to accurately reproduce this structure. The PACMAN bunches in the beginning and in the end of each 144 bunch train have different closed orbits due to the different number of beam-beam long-range encounters. The structure in the horizontal plane is predominately generated by long-range encounters in CMS (horizontal crossing), while the vertical structure are due to long-range encounters in AT-LAS (vertical crossing).
Also, the SuperPACMAN trains missing head-on collisions in ALICE or LHCb are offset in closed orbit with respect to the trains colliding in all experiments. This is due to the use of luminosity levelling by separation in these experiments: the partially separated beams cause a coherent beam-beam kick, which the SuperPACMAN bunches miss at either the ALICE or LHCb experiment.
The TRAIN simulation, which takes into account these particularities of the filling scheme and the partial separation in ALICE and LHCb, agrees well with the measured separation in both planes, and all expected features are well reconstructed.

IX. CONCLUSIONS
Emittance scans were used in LHC proton physics operation since 2015 as a complementary, independent tool for measuring the bunch-by-bunch transverse emittances and closed orbit differences in collisions. One scan pair (horizontal and vertical plane) has been done regularly at the start of collisions and before programmed dumps at the CMS exper- iment. With the operational parameters established, one scan consists of 7 points at different transverse separation, and the full measurement takes less than 5 min.
With the beam and machine parameters used since 2016, the present approach of measuring the bunch length with a full-width half-maximum (FWHM) algorithm and then assuming all planes to be Gaussian lead to systematic errors of up to ∼30 % in the crossing plane. To mitigate this, we propose an analysis to take into account the measured longitudinal profiles. This new approach has been validated in simulations and machine studies. Alternatively, if a longitudinal bunch profile measurement is not available, numeric simulations can be used to derive correction constants for the FWHM analysis. For reproducible machine conditions, such a correction can reduce the uncertainty due to the longitudinal profile down to ∼5 %.
The transverse emittances measured through emittance scans have been compared to measurements from the synchrotron light monitors and the expected values from the AT-LAS and CMS luminosities. An agreement better than 20% in convoluted emittance was found over the full run periods of 2015 and 2016. At a plane-by-plane, bunch-by-bunch level, the agreement between the synchrotron light monitors and the emittance scans was found to be better than 10%. The uncertainty is dominated by systematic effects which affect the absolute emittance scales for all bunches; the individual, relative bunch-by-bunch measurement is more accurate. In 2016 and 2017 LHC operation, the emittance scan measurements have contributed to noticing and correcting calibration and degradation issues on operational emittance measurement (the synchrotron light monitors).
The bunch-by-bunch closed orbit differences observed during the scans are fully explained by beam-beam long-range and PACMAN effects. All bunch-by-bunch features and structures agree with simulation results from the TRAIN code. It is to be noted that over the course of this study, it was found that for the 2017 "ATS" LHC optic [26] the long-range beambeam encounters beyond 15 in the drift space and inner triplet have a significant effect, in particular in the horizontal plane. This highlights the importance of the emittance scan data for benchmarking and improving the simulation code which can then be applied to different scenarios and machines.
Over the course of 2017 LHC operation, the data from emittance scans was also be used by the CMS experiment to continuously check the stability of their individual luminosity monitors over the course of the LHC proton physics runs [28]. Following these results, the ATLAS collaboration requested emittance scans to be commissioned at their experiment as well for 2018. Since the beam sizes at ATLAS and CMS are the same but the beams cross in different planes, comparing the scans at the two experiments will allow a direct measurement of the impact of the longitudinal distribution in the crossing planes.